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An ion trap confines an ion using electromagnetic fields, allowing repeated interrogation of the same particle. The name also covers related traps for an Electron. A Paul trap uses an oscillating electric quadrupole field; a Penning trap combines a static electric quadrupole field with a static magnetic field. Confinement creates time to measure an energy level, frequency or magnetic dipole moment; it does not itself cool the particle or identify its internal quantum state.

The obstacle is Earnshaw's theorem. In a charge-free neighbourhood the static electric potential $\Phi$ obeys Laplace's equation, ${\nabla }^2\Phi =0$. For a particle of fixed nonzero electric charge $Q$, the potential energy $Q\Phi$ is also a harmonic function. The maximum principle for harmonic functions excludes a strict interior minimum. For a quadratic example, the Hessian matrix has zero trace; it cannot have three positive eigenvalues. A static purely electric field therefore cannot supply three-dimensional stable confinement of a free point electric charge. Time variation in a Paul trap, and the velocity-dependent Lorentz force in a Penning trap, change the assumptions of this argument.

Specify the convention before quoting a stability parameter. Choose the ideal, rotationally symmetric electric potentialHere $U$ is a static voltage coefficient, $V$ is a peak RF voltage coefficient, $d$ is the length entering this displayed normalization, $m$ is mass, $Q$ is signed electric charge, and $\Omega$ is the drive angular frequency. The coefficient $V$ is neither a peak-to-peak voltage nor an RMS voltage. An apparatus's ring-to-cap voltage requires matching its electrode surfaces to this electric potential; electrode radius alone does not define our $d$. The minus sign fixes the RF phase at $t=0$.

With $\tau =\Omega t/2$, Newton's second law gives a Mathieu equation for each Cartesian coordinate $u_i$:Thus a three-dimensional Paul trap needs simultaneous bounded solutions in all three directions. Changing the sign of $V$ reverses $q_i$ and shifts the RF phase by half a cycle; it does not change the ideal infinite-time stability regions. Rescaling the voltage or length convention changes the numerical parameters. Floquet theory determines stability of the full Mathieu equation; the small-parameter approximation below does not establish stability for arbitrary $q_i$. Bounded mathematical motion can still hit an electrode if its amplitude exceeds the available space.

The slow part of a trapped trajectory in the small-parameter regime of a Paul trap. Its angular frequency is approximately set by the curvature of the RF-averaged trapping potential, including any static electric potential contribution. Paul trap micromotion is superposed on this slow motion.

In 1995, Christopher Monroe and collaborators cooled one ${}^9{\mathrm{Be}}^{+}$ ion in a ring/endcap Paul trap. They used $313\mathrm{nm}$ Doppler cooling and optical pumping into $|F,m_F⟩=|2,2⟩$. Red-sideband Raman transitions transferred population to $|1,1⟩$, followed by dissipative recycling. Repeating across three directions yielded a reported three-dimensional ground state population of about $92\%$.^[ref]

A variable-frequency Raman transition probe, an exchange pulse and a cycling detection beam converted state populations into fluorescence collected through $f/2$ optics onto a position-sensitive photomultiplier. Red/blue asymmetry inferred motional occupation; varying probe duration tested the thermal assumption. Off-resonant processes and independently observed heating limited cooling.^[ref]

Trapped-ion quantum control prepares an ion's internal quantum state, couples it to quantized motion with laser pulses, and measures the result through state-dependent fluorescence. Its appeal is access to both a controllable two-level system and a quantum harmonic oscillator in one apparatus.

Confinement, preparation, control and readout solve different problems. The Paul trap's RF electric field confines the ion through its RF-averaged trapping potential; it still produces Paul trap micromotion. Doppler cooling and resolved sideband cooling reduce motional excitation. Optical pumping selects an internal quantum state. A coherent pulse then changes amplitudes, while electron shelving converts an internal population into a detectable stream of photons. A bright/dark count by itself does not reveal a quantum superposition's relative phase; an additional controlled rotation before detection can make that phase affect the measured population.